Higher-Order Unconstrained Binary Optimization

• HUBO is a generalization of QUBO that incorporates polynomial functions of order greater than two to model complex combinatorial interactions.
• It provides a more accurate representation of real-world optimization problems in domains like quantum computing, machine learning, and computational chemistry.
• Innovative Ising-space quadratization reduces computational resources by compressing variable and term counts, enabling efficient embedding on resource-constrained hardware.

Higher-Order Unconstrained Binary Optimization (HUBO) generalizes quadratic unconstrained binary optimization (QUBO) by incorporating objective functions that are polynomials in binary variables of order greater than two. These models naturally arise in complex combinatorial optimization problems in domains including quantum computing, statistical physics, wireless communications, finance, machine learning, and computational chemistry. The transition from QUBO to HUBO enables more faithful modeling of multi-variable interactions but presents substantial challenges for efficient encoding, solution, and hardware realization, particularly in resource-constrained or sparsity-sensitive settings.

1. Mathematical Structure and Model Definition

A HUBO problem seeks to minimize (or maximize) a function $f(z)$ comprising polynomial terms of arbitrary degree over a binary domain: $$f(z) = \sum_{S \subseteq [n]} \alpha_S \prod_{i \in S} z_i,$$ where each $z_i$ is a binary variable, and %%%%2%%%% are real-valued coefficients. In Ising notation ($\{-1,1\}$): $$f(s) = \sum_{S \subseteq [n]} \alpha_S \prod_{i \in S} s_i.$$

HUBO captures all multilinear pseudo-Boolean functions, thus encompassing general combinatorial landscapes, including those not representable by quadratic forms.

2. Degree Reduction and Quadratization: Theoretical Foundations and Algorithms

The standard approach to rendering HUBO amenable for quantum annealing and related solvers is "quadratization," i.e., degree reduction to QUBO. The Rosenberg polynomial provides an efficient Boolean-space reduction by introducing an auxiliary variable for every reduction of a single term's degree by one, yielding a quadratic penalty function. However, this is directly applicable only in Boolean ($\{0,1\}^n$) space: $h(x_1, x_2, y) = 3y + x_1x_2 - 2x_1y - 2x_2y,$ which enforces $y = x_1 x_2$.

For Ising variables ($\{-1,1\}$), this method cannot enforce $y = s_1 s_2$ using only one auxiliary variable quadratically; the paper rigorously proves such a quadratic penalty function is impossible in Ising space with a single auxilliary. Applying the Boolean-space method to Ising problems requires variable mapping, resulting in a sparsity blowup—sparse Ising monomials become dense Boolean polynomials, with an exponential number of terms and required auxiliary variables, rendering large instances infeasible on available hardware.

To address this, the direct Ising-space quadratization method introduces two auxiliary variables per reduction. The key quadratic penalty is: $$h(s_1, s_2, y, d) = 4 + s_1 + s_2 - y - 2d + s_1 s_2 - s_1 y - s_2 y - 2 s_1 d - 2 s_2 d + 2 y d,$$ which is zero iff $y = s_1 s_2$ (for some $d$), positive otherwise.

For arbitrary-degree monomials in Ising HUBOs, repeated application of this construction reduces high-degree terms to quadratic, each application introducing two auxiliary variables, resulting in auxiliary variable growth linear in degree ($O(n)$), vastly improving over the exponential scaling of prior approaches.

Two algorithmic heuristics for term breakdown are described:

• Greedy Coverage Algorithm: At each step, select the quadratic reduction covering the largest number of as-yet-unreduced monomials.
• Weighted Bipartite Matching Algorithm: Model terms and dependencies as bipartite graphs, greedily select quadratic reductions maximizing edge weights.

The compressed quadratization approach maintains sparsity, ensuring the QUBO remains as compact as possible.

3. Computational Resource Implications and Empirical Metrics

When converting sparse higher-order Ising problems to QUBO using direct Ising quadratization, the method retains original term sparsity and introduces substantially fewer variables and terms. For instance, on dataset D30C:

• Ising-space quadratization: 706 variables, 3,230 terms
• Boolean + Rosenberg mapping: 1,478 variables, 28,103 terms

This order-of-magnitude compression is crucial for mapping realistic problems onto contemporary quantum annealing or quantum-inspired hardware, which are severely limited in terms of physical variables and interaction graph density.

4. Impact on Quantum and Quantum-Inspired Hardware

Modern quantum annealers and quantum-inspired digital annealers (e.g., D-Wave, Fujitsu Digital Annealer) are designed for QUBO or Ising-type Hamiltonians, but hardware constraints on variable number and interaction density restrict problem sizes. The compressed quadratization method allows larger, more complex HUBO (particularly sparse Ising) problems to be embedded and solved, increasing the practical tractability of applications in chemistry (molecular Hamiltonians), hypergraph problems, MAX-SAT, and related areas.

By reducing variable and coupler count while preserving problem structure and sparsity, the method enables more efficient hardware utilization, supports the solution of greater instance sizes, and preserves the theoretical and empirical optimality of the original HUBO instance.

5. Theoretical Guarantees and Formalization

Quadratization, in general, is formally defined as follows. For a function $f(z)$, a function $h(z, y)$ is a quadratization if: $$f(z) = \min_{y \in \mathcal{B}^m} h(z, y) \quad \forall z \in \mathcal{B}^n$$ where $\mathcal{B} = \{0,1\}$ or $\{-1,1\}$. The direct Ising-space approach provides the first rigorous construct for quadratization in Ising variables, avoiding dense polynomial blowup.

6. Performance Trade-Offs and Summary Table

A direct comparison summarizes the main approaches:

Approach	Space	Aux. variables (single term, degree $n$)	QUBO density	Feasibility on hardware
Rosenberg + Boolean map	Boolean	$O(2^n)$	Dense	Impractical
Direct Ising (this paper)	Ising	$O(n)$	Sparse	Feasible/Scalable

Empirical evidence shows an up to 10× reduction in both variable and term count compared to Boolean-centric strategies, with scalability retained for sparse high-order problems.

7. Applications and Outlook

The ability to quadratize sparse high-order Ising space optimization problems efficiently is critical for advancing quantum optimization in resource-limited environments. Sectors such as quantum chemistry, NP-hard combinatorial optimization, and academia benefit directly from the enhanced ability to model and solve structurally-sparse but high-degree objective functions. The method catalyzes further research into hybrid algorithms, hardware-aware optimization workflows, and foundations of constraint encoding in quantum computing. The formal impossibility results highlight key differences between Boolean and Ising spaces, guiding future protocol design.

In summary, the introduction of compressed quadratization in Ising space fundamentally restructures the landscape for encoding, embedding, and solving HUBO problems, offering a scalable, hardware-adapted, and mathematically concrete foundation for continued combinatorial optimization advances on quantum and quantum-inspired platforms.